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The mean value theorem of calculus states that, given a differentiable function f on an interval [ a , b ] , there exists at least one mean value abscissa c such that the slope of the tangent line at (c , f (c)) is equal to the slope of the secant line through (a , f (a)) and (b , f (b)) . In this article, we study how the choices of c relate to varying the right endpoint b. In particular, we ask: When we can write c as a continuous function of b in some interval? As we explore this question, we touch on the implicit function theorem, a simplified version of the Morse lemma, and the theory of analytic functions. [ABSTRACT FROM AUTHOR]

We investigate functions with the property that for every interval, the slope at the midpoint of the interval is the same as the average slope. More generally, we find functions whose average slopes over intervals are given by the slope at a weighted average of the endpoints of those intervals. This is equivalent to finding functions satisfying a weighted mean value property. In the course of our exploration, we find connections to harmonic functions that prompt us to explore multivariable analogs and the existence of "weighted harmonic functions". [ABSTRACT FROM AUTHOR]